1.

Find the square root of the following complex number: `-5+12 i`

Answer» Let `sqrt(-5+12i)=(x+iy)." "...(i)`
On squaring both sides of (i), we get
`-5+12i=(x+iy)^(2) rArr -5+12i=(x^(2)-y^(2))+i(2xy)." "...(ii)`
On comparing real parts and imaginary parts on both sides of (ii), we get `x^(2)-y^(2)=-5 and 2xy = 12`
`rArr" "x^(2)-y^(2)=-5 and xy = 6`
`rArr" "(x^(2)+y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt((-5)^(2)+4xx36)=sqrt(169)=13`
`rArr" "x^(2)-y^(2)=-5 and x^(2)+y^(2)=13`
`rArr" "2x^(2)=8 and 2y^(2)=18`
`x^(2)=4 and y^(2)=9`
`rArr" "x = +- 2 and y = +- 3`.
Since `xy gt 0`, so x and y are of the same sign.
`therefore" "(x=2 and y=3) or (x =-2 and y =-3)`.
Hence, `sqrt(-5+12i) = (2 + 3i) or (-2 -3i)`.


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